Quantum phenomenology from a classical integer automaton: an emergent Schrödinger sector, exact exchange statistics, CHSH violation without randomness, foliation-independent Bell statistics, a selection-rule theorem for ensemble entanglement, a stimulated parametric pair channel with a capacity wall, a surviving isotropy no-go, two engine-measured forced rationals, and obstruction theorems for dynamic geometry

Abstract

What this is. A single bit-exactly reversible, integer-valued lattice automaton in which a broad range of quantum phenomenology arises as measurable, checksum-reproducible behaviour. Evolution is a permutation of integer microstates, not approximate numerical integration — which is why the gates below can be bit-exact rather than statistical. This is an existence result strengthened by a survived no-go and one falsifiable prediction; it is not a claim that our universe is this automaton. Quantum sector. An emergent Schrödinger sector (envelope bridge with effective mass fixed by the curvature of the exact discrete dispersion; two microscopically different substrates collapsing onto the single universal spreading law √(1+τ²)); deterministic double-slit single-object interference (p<10⁻⁴) with trajectories matching the weak-measurement reconstruction of Kocsis et al. 2011; CHSH = 2.79 from position-valued outcomes with a no-signalling control; a measured branch-coherence decay law (R²=0.997); exact exchange statistics, with Pauli exclusion holding as a bit-conserved invariant for all time; a quantum eraser with literal bit-reversible un-measurement; and partial coarse-grained Born relaxation with exact Loschmidt reversal. Where the quantum–classical boundary lies. The two-particle interaction is substrate-derived (cross-/self-phase ratio 2.31 against a parameter-free prediction of 2), so both terms of the two-particle Hamiltonian are emergent — but the reconstructed state stays Schmidt rank 1 where the L² reference reaches K=4.9. The substrate derives the Hamiltonian, not the L² state space; entanglement is carried, not derived. Spin is implemented. Lorentz structure. Velocity anisotropy vanishes as k² in the infrared with first-principles coefficients in 2D and 3D (≈1%), so rotational isotropy emerges as an infrared fixed point. Bell statistics are foliation-order-independent for causally separated measurement operations (|ΔS|<0.005, S>2 in every ordering) while 41% of individual trajectory pairs flip — a preferred foliation that trajectories know about and no measurement reveals, with the residual order-dependence scaling away as A⁻⁰⋅⁹³. Full boost covariance of states and fields remains open. New in v4 — a dynamic-geometry sector, reported with its obstructions. Six pre-registered pilots ask whether matter can make the local-time geometry and be acted on by it. Three results are derived and measured: (i) an exactly conserved geometric ledger forces the static response to be contact-only — the lattice counterpart of Gauss’s law on a compact space — so no two-body force is possible in that class (hypotheses stated explicitly; this is a no-go for a class, not for discrete gravity in general); (ii) a source entering the first-order equation yields a flat plateau in 1D (depth 46.3 against the parameter-free 45.25, constant in time, exactly zero outside the causal cone), 1/t dilution in 2D and nothing in 3D, so transport cannot build a static sourced field in any dimension; (iii) four independently probed force channels are blocked, the only directional one carrying the winding (electromagnetic) sign. What the ledger does permit. Relaxing conservation of the free field while keeping the total exact restores long range without losing bit-exactness (far/source 0.98–1.06 against ≈0 for contact), and a carry-based binding rule makes mass = number of bound tokens a 1% measurement rather than a framing (rate drift 1.3% while the object’s volume changed 3.05×). The substance carries a second bitwise-exact Gauss law (div e − ΔQ = 0 over 2×10⁴ ticks, flux = enclosed charge exactly) whose charge is the conserved energy. Letting the clocks read it closes the matter→geometry→matter loop kinematically: a body slows its own clock by 30%, matching the clock-rate law to 1.3%, with a causal onset. In 2D the clock field refracts a mobile probe, and a binary, untunable band-edge prediction tabulated before the run is confirmed at both clock ratios (transmission 0.008–0.046 in the predicted reflecting region against 0.58–0.93 just below it); the quantitative refraction law is closed as unproven after two attempts, per its own pre-registration. None of this is general relativity: the geometry is a scalar lapse, no attraction is measured, and the obstruction theorem forbids one in the conserving class. Falsifiable prediction, factor-corrected in v4. The quadratically Planck-suppressed photon dispersion coefficient is measured on the engine rather than derived from a formula: ξiso(3D) = 0.024299 ± 1.1×10⁻⁴ against the exact rational 187/7680 (0.20%), with the subluminal sign resolved from zero by 221 numerical standard errors of the fit. Three coefficients are now stated separately: phase (ξ), group velocity (3ξ) and the Lang convention, δγ,2 = −2ξ/EPl² = −3.27×10⁻⁴⁰ GeV⁻². Our earlier headline quoted the phase coefficient under the Lang symbol; that factor of 2 is the nineteenth self-correction on record and was found by an external automated review, credited as such. The prediction sits ~30× below the older model-dependent Lang et al. (2017) bound, but is in tension (~3.3×) with the conditional Pierre Auger (2022) limit for a source scenario with a subdominant proton component — so it is now decidable by UHECR composition measurements, with a threshold signature at E*≈2 EeV. Its fate depends on the proton fraction at the highest energies, the source model, and the assumption of standard interaction-vertex kinematics, which this substrate does not derive. New in v5 — a selection-rule theorem, a stimulated pair channel, a capacity wall, and a second forced rational. The two-particle boundary is turned into a theorem, attacked adversarially, and mapped constructively. An exact selection rule for the pre-registered phase-ensemble reconstruction (the e−iφ weight pins harmonic pairs to n+m=1) forces any U(1)-equivariant dynamics to factorize exactly — Schmidt rank 1 regardless of interaction strength — with the Schmidt excess bounded at fourth order in the equivariance defects (a one-sided defect buys nothing); the proof passed an adversarial proof-check. A red-team scan of six rank-maximizing mechanisms returned NULL in the autonomous class, and the sharpest impostor — a Floquet drive that initially passed every discriminator — was convicted by the scaling of its rank signal with the integer quantization step itself (423×): the twentieth self-correction, retracted the day it was born. Constructively, a coherent pump prepared in the initial condition of the time-homogeneous update converts through the independently measured cubic coefficient into stimulated, phase-sensitive correlated envelope pairs in two exactly momentum-matched channels — one an umklapp channel whose recoil G = 2π is supplied by the lattice itself — with a parameter-free coupling κ = γP, gain locked to twice the pump phase and scaling as A², and a Manley–Rowe ledger (pump depletion = 2× pairs) closing at the percent level. A capacity scan across pump power, interaction time, conversion-zone length and channel number finds the Schmidt number saturated at the pair (K ≤ 1.12, never more than 2 significant modes, ~65× below the contact L² reference): the substrate makes pairs, not volume — the tensor-product wall stands. A theorem-backed classification fixes where Bell violation can live: configuration-space dynamics realizes it by construction (the earlier S = 2.79 is reread as a confirmation of that class, not an empirical surprise), while for the physically local pumped class per-run CHSH ≤ 2 is a theorem — the driven correlations are classical entanglement: nonseparability without nonlocality. One order deeper in the dispersion, the lattice forces a second exact rational: <ξ₂>3D = 103351/495452160, pre-registered and engine-confirmed to 0.31% (S/N 312), with the first-allowed cos 8θ harmonic detected at S/N 29 at its forced amplitude −1/36864 — a structural forced constant ~56 orders below current bounds, stated as such, not a new testable prediction. A closing section translates the results into renormalization-group vocabulary (isotropy as an IR fixed point; a two-substrate universality class; quantization-floor artefacts as lattice-scale operators) — a dictionary, explicitly not a computed Wilsonian flow. The archive adds the full pre-registrations, engines and per-run JSONs for all four campaigns (~$10 of cloud compute). Methodology as a co-equal contribution. Criteria are committed to version control before each run; the commit hash is the timestamp. Twenty headline results have been retracted or corrected when the project’s own tests exposed artefacts — five in this version. The archive ships the pre-registration designs themselves (specs/), including the correction blocks for changes that failed: a coupling measured to pump energy and reported as “measured, not validated”; a gate left failed on a statistic carrying 0.4% of the signal power; a predicted field inversion that did not occur; and three measuring instruments retired after being shown ill-posed. v4 also discloses and fixes a reproducibility defect in the v3 archive: argparse defaults differed from the published configuration and one report JSON was omitted, so exact command lines are now given verbatim in the README. Prior art. Individual phenomena have known precedents — Philippidis–Dewdney–Hiley 1979 (double-slit Bohmian trajectories); Valentini–Westman 2005 (Born relaxation); Visscher 1991 (the update scheme), with the integer/floor-reversible lifting already noted by Fredkin 1999 and Martin-Delgado 2004; Dürr et al. 1999 (a preferred foliation hidden from equilibrium statistics) — and are framed accordingly. The contribution is their tested, validated realization on one exactly reproducible integer substrate, several new exact constructions, the obstruction results above, and the pre-registered, falsification-first methodology. Produced in an AI-assisted workflow (Anthropic Claude) under th